Real-rooted atlas

A038337 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(45 + 22\,x + 22\,n\right)\,P_{n}(x) = \left(1541 + 1980\,x + 484\,x^{2} + 506\,n + 484\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1541 + 1980\,x + 484\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0154119804844005
1470811017066534010648224775
2135053741431284475064191664023425612119625
337218853148559070227872040164085680527076005153632635596875
499763316549128885969804898620100263697605414827440139148064011337990432661478125
5262023523531536355074103782924218925033181291603860152357601667766851523571466976024943578881652449359375
667756026395346116140141281351992465208022193155417664221459976830401358773629875248921160977928979688396805487587353682580949140625
71730672774710113415693226269445654978739867289231502703728010985611931743688769815082483844483952978588161383769981946882222472878208012072692177924086075469921875

All 15 rows

Technical details
Node id
n-8ad9fc69d69d537e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-9f722f4116b0c75f

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:              1541,               1980,                484;
  1:             47081,             101706,              65340,               10648;
  2:           1350537,            4143128,            4475064,             1916640,              234256;
  3:          37218853,          148559070,          227872040,           164085680,            52707600,             5153632;
  4:         997633165,         4912888596,         9804898620,         10026369760,          5414827440,          1391480640,           113379904;
  5:       26202352353,       153635507410,       378292421892,        503318129160,        386015235760,        166776685152,         35714669760,          2494357888;
  6:      677560263953,      4611614014128,     13519924652080,      22193155417664,      22145997683040,      13587736298752,       4892116097792,        897968839680,        54875873536;
  7:    17306727747101,    134156932262694,    456549787398672,     892315027037280,    1098561193174368,     876981508248384,     448395297858816,     138376998194688,     22224728782080,      1207269217792;
  8:   437680096735509,   3807480104362220,  14757262548896340,   33480317742569280,   49077326487050400,   48336692499672192,   32155988635774080,   14092423646991360,   3805367450353920,    543271148006400,    26559922791424;
  9: 10978025471616985, 105918583409993178, 460705092627828620, 1190419178944304760, 2025559223425441440, 2375342601973239360, 1949579930820111744, 1111678464265332480, 426295815321488640, 102322102553960960, 13147161781754880, 584318301411328;
%F Formula
The row polynomials satisfy (22*n+45+22*x)*P(n,x) = (506*n+1541+(484*n+1980)*x+484*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1541 + 1980*x + 484*x^2.
%t Mathematica
P[0, x_] := 1541 + 1980*x + 484*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((506*n + 1541 + (484*n + 1980)*x + 484*x^2)*P[n - 1, x])/(22*n + 45 + 22*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1541 + 1980*x + 484*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((506*n + 1541 + (484*n + 1980)*x + 484*x^2)*v[i - 1])/(22*n + 45 + 22*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.